Yes, you can build a PC for dirt, dirt cheap. However, how cheap is too cheap? Gizmodo breaks down exactly what you get for your money, with the help of Maximum PC's Will Smith. Check out the price breakdowns below- you can actually build a functioning PC (essentially a nettop) for just $131.
Smith recommends you go for the $647 model, and (bonus!) he posted a really great HowTo on how to build it.